Problem 140 of Project Euler is very much a continuation of the Problem 137, as we can see from the problem description
Consider the infinite polynomial series AG(x) = xG1 + x2G2 + x3G3 + …, where Gk is the kth term of the second order recurrence relation Gk = Gk-1 + Gk-2, G1 = 1 and G2 = 4; that is, 1, 4, 5, 9, 14, 23, … .
We shall call AG(x) a golden nugget if x is rational. Find the sum of the first thirty golden nuggets.
In Problem 137 I mentioned in the end that the problem could be solved using a Diophantine equation. This is exactly the way I will go for this problem.